79,584 research outputs found

    A bandwidth theorem for approximate decompositions

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    We provide a degree condition on a regular nn-vertex graph GG which ensures the existence of a near optimal packing of any family H\mathcal H of bounded degree nn-vertex kk-chromatic separable graphs into GG. In general, this degree condition is best possible. Here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size. Equivalently, the separability condition can be replaced by that of having small bandwidth. Thus our result can be viewed as a version of the bandwidth theorem of B\"ottcher, Schacht and Taraz in the setting of approximate decompositions. More precisely, let δk\delta_k be the infimum over all δ1/2\delta\ge 1/2 ensuring an approximate KkK_k-decomposition of any sufficiently large regular nn-vertex graph GG of degree at least δn\delta n. Now suppose that GG is an nn-vertex graph which is close to rr-regular for some r(δk+o(1))nr \ge (\delta_k+o(1))n and suppose that H1,,HtH_1,\dots,H_t is a sequence of bounded degree nn-vertex kk-chromatic separable graphs with ie(Hi)(1o(1))e(G)\sum_i e(H_i) \le (1-o(1))e(G). We show that there is an edge-disjoint packing of H1,,HtH_1,\dots,H_t into GG. If the HiH_i are bipartite, then r(1/2+o(1))nr\geq (1/2+o(1))n is sufficient. In particular, this yields an approximate version of the tree packing conjecture in the setting of regular host graphs GG of high degree. Similarly, our result implies approximate versions of the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree.Comment: Final version, to appear in the Proceedings of the London Mathematical Societ

    A combinatorial approach to optimal designs.

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    PhDA typical problem in experimental design theory is to find a block design in a class that is optimal with respect to some criteria, which are usually convex functions of the Laplacian eigenvalues. Although this question has a statistical background, there are overlaps with graph and design theory: some of the optimality criteria correspond to graph properties and designs considered ‘nice’ by combinatorialists are often optimal. In this thesis we investigate this connection from a combinatorial point of view. We extend a result on optimality of some generalized polygons, in particular the generalized hexagon and octagon, to a third optimality criterion. The E-criterion is equivalent with the graph theoretical problem of maximizing the algebraic connectivity. We give a new upper bound for regular graphs and characterize a class of E-optimal regular graph designs (RGDs). We then study generalized hexagons as block designs and prove some properties of the eigenvalues of the designs in that class. Proceeding to higher-dimensional geometries, we look at projective spaces and find optimal designs among two-dimensional substructures. Some new properties of Grassmann graphs are proved. Stepping away from the background of geometries, we study graphs obtained from optimal graphs by deleting one or several edges. This chapter highlights the currently available methods to compare graphs on the A- and D-criteria. The last chapter is devoted to designs to which a number of blocks are added. Cheng showed that RGDs are A- and D-optimal if the number of blocks is large enough for which we give a bound and characterize the best RGDs in terms of their underlying graphs. We then present the results of an exhaustive computer search for optimal RGDs for up to 18 points. The search produced examples supporting several open conjectures

    Mars: Near-Optimal Throughput with Shallow Buffers in Reconfigurable Datacenter Networks

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    The performance of large-scale computing systems often critically depends on high-performance communication networks. Dynamically reconfigurable topologies, e.g., based on optical circuit switches, are emerging as an innovative new technology to deal with the explosive growth of datacenter traffic. Specifically, periodic reconfigurable datacenter networks (RDCNs) such as RotorNet (SIGCOMM 2017), Opera (NSDI 2020) and Sirius (SIGCOMM 2020) have been shown to provide high throughput, by emulating a complete graph through fast periodic circuit switch scheduling. However, to achieve such a high throughput, existing reconfigurable network designs pay a high price: in terms of potentially high delays, but also, as we show as a first contribution in this paper, in terms of the high buffer requirements. In particular, we show that under buffer constraints, emulating the high-throughput complete-graph is infeasible at scale, and we uncover a spectrum of unvisited and attractive alternative RDCNs, which emulate regular graphs of lower node degree. We present Mars, a periodic reconfigurable topology which emulates a dd-regular graph with near-optimal throughput. In particular, we systematically analyze how the degree dd can be optimized for throughput given the available buffer and delay tolerance of the datacenter

    A Comprehensive Methodology for Algorithm Characterization, Regularization and Mapping Into Optimal VLSI Arrays.

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    This dissertation provides a fairly comprehensive treatment of a broad class of algorithms as it pertains to systolic implementation. We describe some formal algorithmic transformations that can be utilized to map regular and some irregular compute-bound algorithms into the best fit time-optimal systolic architectures. The resulted architectures can be one-dimensional, two-dimensional, three-dimensional or nonplanar. The methodology detailed in the dissertation employs, like other methods, the concept of dependence vector to order, in space and time, the index points representing the algorithm. However, by differentiating between two types of dependence vectors, the ordering procedure is allowed to be flexible and time optimal. Furthermore, unlike other methodologies, the approach reported here does not put constraints on the topology or dimensionality of the target architecture. The ordered index points are represented by nodes in a diagram called Systolic Precedence Diagram (SPD). The SPD is a form of precedence graph that takes into account the systolic operation requirements of strictly local communications and regular data flow. Therefore, any algorithm with variable dependence vectors has to be transformed into a regular indexed set of computations with local dependencies. This can be done by replacing variable dependence vectors with sets of fixed dependence vectors. The SPD is transformed into an acyclic, labeled, directed graph called the Systolic Directed Graph (SDG). The SDG models the data flow as well as the timing for the execution of the given algorithm on a time-optimal array. The target architectures are obtained by projecting the SDG along defined directions. If more than one valid projection direction exists, different designs are obtained. The resulting architectures are then evaluated to determine if an improvement in the performance can be achieved by increasing PE fan-out. If so, the methodology provides the corresponding systolic implementation. By employing a new graph transformation, the SDG is manipulated so that it can be mapped into fixed-size and fixed-depth multi-linear arrays. The latter is a new concept of systolic arrays that is adaptable to changes in the state of technology. It promises a bonded clock skew, higher throughput and better performance than the linear implementation

    Group testing with Random Pools: Phase Transitions and Optimal Strategy

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    The problem of Group Testing is to identify defective items out of a set of objects by means of pool queries of the form "Does the pool contain at least a defective?". The aim is of course to perform detection with the fewest possible queries, a problem which has relevant practical applications in different fields including molecular biology and computer science. Here we study GT in the probabilistic setting focusing on the regime of small defective probability and large number of objects, p0p \to 0 and NN \to \infty. We construct and analyze one-stage algorithms for which we establish the occurrence of a non-detection/detection phase transition resulting in a sharp threshold, Mˉ\bar M, for the number of tests. By optimizing the pool design we construct algorithms whose detection threshold follows the optimal scaling MˉNplogp\bar M\propto Np|\log p|. Then we consider two-stages algorithms and analyze their performance for different choices of the first stage pools. In particular, via a proper random choice of the pools, we construct algorithms which attain the optimal value (previously determined in Ref. [16]) for the mean number of tests required for complete detection. We finally discuss the optimal pool design in the case of finite pp
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